Page 92 - J. C. Turner - History and Science of Knots

The Peruvian Quipu 81

on them that might indicate that these knots did not represent numerals. For
example, Nordenskiold describes ten- to fifteen-fold long knots ([11], p. 18)
and long knots which appear in the tens' position ([12], p. 16).
A second, more refined possibility for noting down non-numerical informa-
tion on a quipu is a label code. There might have been a system of translating
spoken words into numbers, like we translate letters into numbers by using
the ASCII code. In this case it would be possible to read a quipu when just
knowing the code, but not its content. Yet there was no possibility of writing
down a translation table. Reading this kind of quipu would have to be learned
from oral instruction, and so the skill, if it existed, vanished with the Inca
people under the Spanish conquest.

Incan Mathematics

The collection of numbers on the ancient quipus allow us to reconstruct some
of the mathematics behind them. In this section, some of the studies in this
field are reviewed. Possible arithmetic operations and geometric interpreta-
tions are discussed. Finally, a calculation device is described which the Incas
might have used.
Arithmetics
From their analysis of the structure of a vast number of quipus , Ascher and
Ascher deduced ([61, pp. 133-155) that the arithmetics used by the Incas must
have included at least:
(1) addition
(2) division into equal parts
(3) division into unequal parts
(4) multiplication of integers by integers and fractions.
That means that the Incas dealt with fractional values in the form of division
into parts and common ratios, though fractions cannot be encoded on the
quipus.
The following examples are to demonstrate the occurrence of the various
arithmetical operations. They are all taken from Code of the Quipu. The qui-
pus are referred to by the labels by which they are catalogued in [5]; the letters
denote the author who first described the quipu, and the number indicates the
order of publication.
Addition: We already saw in the section about the spatial layout that top
cords or extra groups carry the sums of the numbers on other cords. This
summation appears on about 25% of the quipus examined by Ascher and
Ascher.